$1.($S$\backslash$&N $15\mathrm{.}2)$ Suppose that firms' marginal and average costs are constant and equal to $\backslash ($ c $\backslash )$ and that inverse market demand is given by $\backslash ($ P$=$a$-$b Q $\backslash ),$ where $\backslash ($ a$,$ b$>0\backslash )$

a$)$ Calculate the profit maximizing price$-$quantity combination for a monopolist. Also calculate the monopolist's profit.

b$)$ Calculate the Nash equilibrium quantities for Cournot duopolists, which choose quantities for their identical products simultaneously. Also compute market output, market price and industry profits.

c$)$ Calculate the Nash equilibrium prices for Bertrand duopolists, which choose prices for their identical products simultaneously. Also compute firm and market output as well as firm and industry profits.

d$)$ Suppose now there are $\backslash ($ n $\backslash )$ identical firms in a Cournot model. Compute the Nash equilibrium quantities as functions of $\backslash ($ n $\backslash ).$ Also compute market output, market price, firm and industry's profits. Show that as $\backslash ($ n $\backslash )$ increases, the market outcomes will move closer to perfect competition.